Non-local distance functions and geometric regularity

TL;DR

This paper links the geometric regularity of sets to the behavior of smooth non-local distance functions, providing new analytic tools for understanding rectifiability across all dimensions without topological constraints.

Contribution

It establishes a novel equivalence between rectifiability and gradient oscillation estimates for non-local distance functions, expanding the analytic characterization of geometric regularity.

Findings

Equivalence between set regularity and gradient oscillation estimates.

Applicable to all dimensions and co-dimensions.

No topological assumptions required.

Abstract

We establish the equivalence between the regularity (rectifiability) of sets and suitable estimates on the oscillation of the gradient for smooth non-local distance functions. A prototypical example of such a distance was introduced, as part of a larger PDE theory, by Guy David, Joseph Feneuil, and the third author. The results apply to all dimensions and co-dimensions, require no underlying topological assumptions, and provide a surprisingly rich class of analytic characterizations of rectifiability.